Deformation of the σ2-curvature.

Our main goal in this work is to deal with results concern to the σ2<inline-graphic></inline-graphic>-curvature. First we find a symmetric 2-tensor canonically associated to the σ2<inline-graphic></inline-graphic>-curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of σ2<inline-graphic></inline-graphic>-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and σ2<inline-graphic></inline-graphic>-curvature. With a suitable condition on the σ2<inline-graphic></inline-graphic>-curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative σ2<inline-graphic></inline-graphic>-curvature unless it is flat. [ABSTRACT FROM AUTHOR]